Advances in Electrical and Computer Engineering Volume 16, Number 1, 2016 [602266]

Advances in Electrical and Computer Engineering Volume 16, Number 1, 2016
A Comparative Parametric Analysis of the
Ground Fault Current Distribution on Overhead
Transmission Lines
Maria VINȚ AN
Lucian Blaga University of Sibiu, Faculty of Engineering, 550025, Sibiu, Romania
[anonimizat]

1Abstract —The ground fault current distribution in an
effectively grounded power network is affected by various
factors, such as: tower footing impedances, spans lengths,
configuration and parameters of overhead ground wires and
power conductors, soil resistivity etc. In this paper, we
comparatively analyze, using different models, the ground fault
current distribution in a single circuit transmission line with
one ground wire. A parametric comparative analysis was done
in order to study the effects of the non-uniformity of the towers
footing impedances, number of power lines spans, soil
resistivity, grounding systems resistances of the terminal
substations etc., on the gro und fault current distribution.
There are presented some useful qualitative and quantitative results obtained through a complex dedicated developed
MATLAB 7.0 program.

Index Terms —fault currents, power system faults,
transmission lines.
I. INTRODUCTION
A ground fault that appears anywhere in a power system
causes fault currents through the grounding systems of all
substations with grounded neutrals. When the fault occurs at any tower of an overhead tran smission line in an effectively
grounded power network, the fault current returns to the grounded neutral through the towers, ground return path and ground wires. The estimation of the ground fault current distribution is an important step to design a safe substation
grounding grid and the associat ed line's grounding systems,
and it had been undertaken by many researches and
numerous analytical methods have been published [1-16]. Rudenberg [2] introduce an an alytical method based on
Kirchoff’s theorems, in order to determine the ground fault
current distribution in effectively grounded power network.
This method did not include the mutual coupling between the ground wire and the faulte d phase. In Endrenyi's [3]
approach, considering a series of identical spans, the tower
impedances and overhead ground wires are reduced to an equivalent lumped impedance. Verma and Mukhedkar [4] included the mutual coupling between the neutral conductors and the phase condu ctors. All the above three
cited works have considered only uniform span lengths and uniform tower resistances, t oo. Sebo [5] introduce an
analytical method which is valid for non-uniform span lengths and non-uniform tower resistances. Goci and Sebo
[6] presented an improved method through the current
distribution is calculated based on the driving point impedance computation. Other methods which permit varying tower resistances alon g a t ransm
ission line have
been described by Dawalibi [7, 8]. These methods were implemented in some complex software frameworks, too.
Popovic introduced an analyti cal procedure which enables
evaluation of the significant parts of the ground fault
current, for a fault at any tower of a transmission line of an
arbitrary number of spans, si ngle or double circuit parallel
lines [9, 10]. His work is focused on the critical fault
position’s determination. In contrast, we focused on the
distribution of the ground fault current in ground wires and return paths. Nahman introduced a mathematical model and equivalent schemes for zero-sequence components for
overhead transmission lines with non-uniform span lengths
and non-uniform tower resistance [11, 12]. This approach is
focused on the effects of voltage drops in the grounding
systems formed by the ground wire and towers, and ground
electrodes of the terminal substations.

1 In our previ ous works
there were presented some
analytical methods in order to determine the ground fault
current distribution in effectively grounded power networks, for a ground fault located anyw here along the transmission
line [17-20]. In all these cases it was assumed uniform spans lengths and towers impedances. But usually these parameters are not the same on the entirely transmission line.
In this paper, we comparatively analyze, using different
models, the ground fault current distribution in a single circuit transmission line with one ground wire. We consider the case when the section of the line between the faulted
tower and the source station is finite, thus it must take into consideration the termination of the network. Also we will treat both cases: uniform, resp ectively non-uniform spans
lengths and towers impedances. Therefore, the new significant contributions of this work comparing with our
previous ones [17-20] are the followings: presenting a
complex analytical model treating the more realistic case of the non-uniform spans lengths and towers impedances; comparing in a quantitative manner, based on some
laborious numerical simulations, this model with the
previous ones, focused on uniform spans lengths and towers impedances.
The calculation methods are based on the following
assumptions: the impedances are lumped parameters in each span of the transmission line; the line’s capacitances are neglected; the contact resistance between the tower and the ground wire, the contact resi stance between the tower and
the faulted phase, are all neglected; the network is
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1582-7445 © 2016 AECE
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Advances in Electrical and Computer Engineering Volume 16, Number 1, 2016
co
nsidered linear in the sinuso idal steady-state, state and
only the fundamental frequency is considered.
II. GROUND FAULT CURRENT DISTRIBUTION
Let's consider a single circuit transmission line with one
ground wire connected to the ground at every tower of the
line, each transmission tower having its own grounding electrode or grid and let’s assume that a single line-to-
ground fault appears at one tower of this transmission line. The fault current returns to the grounded neutral through the ground wires, towers and ground return paths. The fault divides the line into two sections, each extending from the fault towards one end of the line. Depending of the number
of towers between the faulted tower and the stations,
respectively of the distance be tween the towers, these two
sections of the line may be considered infinite or they may be regarded as finite. In the first case, ground fault current
distribution is independent on the termination of the
network. In the second case, the ground fault current distribution may significantly depend on the termination of the network [3].
On the other hand, the ground fault current distribution is
affected by the towers and ground wire impedances. It may be possible that considering uniform spans lengths and towers resistances to result in wrong values because usually,
these parameters are not the same on the entirely
transmission line.
A. Model 1. Uniform Spans Lengths and Tower Resistances
As a first step and without loosi
ng the approach’s
generality, it is assumed that the fault occurs at the last
tower (tower no. 0 in Fig. 1). It is considered that the
transmission line has N towers. Fig. 1 presents the
connection of a ground wire to earth through transmission towers. It is assumed that all the transmission towers have
the same ground impedance and the distance between
to
wers is long enough to avoid the influence between their
grounding electrodes. The self-impedance of the ground
wire connected between two grounded towers, called the
self-impedance per span, was noted with . It was
assum
ed that the distance between two consecutive towers is
the same for every span. represents the mutual-
im
pedance between the ground wire and the faulted phase
conductor, per span. The st ation C grounding system
resistance is . tZ
mZwZ
sR

Figure 1. Ground fault current distribution in case of a fault at the last tower
of the transmission line

As it was already presented in [17-20], considering the n-
th tower, as counted from the terminal tower, the current flowing to ground through it, has an exponential variation
and it is given by the next expression [2], [4]:
nBenAetnI  (1)
A and B in expression (1) are arbitrary parameters and
they could be obtained from the boundary conditions;
parameter  in the solution (1) is given by the next
expression:
tZwZ
 (2)
The current in the ground wire is given by the following
expression:
pIwnIeeBeeAn n
 


1 1 (3)
 in expression (3) represents the coupling factor
between the overhead phase and ground wire (
wm
ZZ )
and represents the fault current. The boundary condition
(co
ndition for n=0) at the terminal tower of Fig. 1 is: pI
0 1tIwIpI  (4),
where represents the current in the faulted tower and
represents the current in the first span of the ground
wire. 0tI
1wI
1) Long Line
If the line is sufficiently long so that, after some distance,
the varying portion of the current exponentially decays to zero, then the parameter . In this case only the
param
eter B must be determined from the boundary
conditions [4]. According to relations (1) and (3), it results: 0A
nBetnI (5)
pI ene BwnI    ) 1 / ( (6)
Substituting these expressions in (4), with n=0 for
and tnI
1n for , it could be obtained the parameter B. wnI
Usually, the terminal tower is connected through an extra
span to the station grounding grid (Fig. 2). As a
con
sequence, the ladder network representing the
transmission line must be closed by a resistance representing
the grounding system of the station resistance. '
wZ
Figure 2. Ground fault current distribution considering the resistance of
station grounding system
sR' in Fig. 2 represents the grounding system resistance
of the station D.
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Advances in Electrical and Computer Engineering Volume 16, Number 1, 2016
109In this case, a part of the total ground fault current will
flow through the station ground resistance . '
sR
wZtZ
n BnBe 6 , 4100 ln
1001   
 (9)
In
order to use the previous results, it is enough to replace
the current with , where pI' '
sIpIpI '
sI represents
the current through the station D grounding grid resistance
and it is given by the next expression:
''
sZ ZZ
pIsI
 (7), For 03 . 0
tZwZ, it will get n =26.5. It takes at least 26
towers from the fault to get a current reduced to 1% then
that traversing the terminal tower.
If the number of the towers is at least equal with the
number given by expression (9), then it is possible to
consider A=0 in the expressions (1) and (3).
whe
re with was noted the sum between and
( ) and '
sZ
'
w'
wZ'
sR
' '
s sZ R Z  Z represents the resultant impedance
of the ladder network looking back from the fault.
In case the values of and '
sZ Z are known, '
sI can be
computed from relation (7).
'
pI is given then by the next expression:
0 1 '' '
tIwI
sZ ZZ
pIpIsIpIpI  
    (8)
Next, it will be found the n-th tower, as counted from the
terminal tower, where the curre nt gets reduced to 1% then
that traversing the te rminal tower [2].
From equation (5) it is obtained: 2) Short Line
If the line cannot be considered long enough regarding to
the expression (9), then parameters A and B will be
determined from the boundary conditions.
The boundary condition at the faulted tower is:
'0 0 1 1 0'
sZtZ
tItIwIwItIsIpI       (10)
At the left terminal of the line we have (see Fig. 2):

   
0' '
1 ,1 ,
wZpIwZN wIsRsItZtNIN wIsIpI
 (11)
Taking into account the ab ove boundary conditions,
according to (1) and (3), parameters A and B will get the
next expressions [7, 18, 19]:

















 
 





 






 


'11)'
1 (
1'11)'
1 (
1'11)'
1 (
1) 1 (
sZtZ
esRtZ
wRwZ
ee Ne
sZtZ
esRtZ
wRwZ
ee NesZtZ
esRtZ
wRwZ
ee NepI
A  
  
(12)


















 
 





 



 
 


'11)'
1 (
1'11)'
1 (
1)'
1 (
1'11) 1 (
sZtZ
esRtZ
wRwZ
ee Ne
sZtZ
esRtZ
wRwZ
ee NesRtZ
wRwZ
ee Ne
sZtZ
epI
B 
 

(13)
B. Model 2. Non-Uniform Spans Lengths and Tower
Resistances
In order to evaluate the ground fault current distribution
in substations, overhead ground wires and towers, a
mathematical model inspired by Sebo’s work [5] is described.
Fig. 3 presents the connection of the ground wire
connected to earth through transmission towers and the
ground fault current distribution when a single line-to-ground fault appears at the last tower.
It is considered that the transmission line has N towers
between the faulted tower and the source station, and that the span lengths and tower resistances are non-
uni
form (k is the number of considered span). ) (kl) (k tZ
The self-im
pedance of the ground wire connected
between two grounded towers, called the self-impedance per
span, was noted with . ) (k wZ
The self-im
pedance of the fa ulted phase conductor per
span was noted with . ) (kpZ
) (k mZ represents the mutual-impedance between the
ground wire and the faulted phase conductor, per span. [Downloaded from www.aece.ro on Tuesday, December 20, 2016 at 18:39:51 (UTC) by 109.101.251.63. Redistribution subject to AECE license or copyright.]

Advances in Electrical and Computer Engineering Volume 16, Number 1, 2016

Figure 3. Fault current distribution on a single circuit transmission line

The source station grounding system resistance is and
represents the grounding system resistance of the
d
istribution station. sR
'
sR
C
onsidering span number k between two consecutive
towers (see Fig. 4), the following expressions, written in a
matrix form, which relate the left-side quantities , , and of the span with its right-side
q
uantities , U , and , can be written
[5]
:
) 1 (k pU) 1 (k wU) 1 (k wI
) (k pU) 1 (k pI
) (k w (wI
[1 (kM)k
])) (k pI
[ ]) (k ] [) (kN E (14)
Or:









  





) () () (
) () (
) ( ) () () ( ) () ( ) (
) 1 () 1 () 1 (
) 1 (101 00 0 1 00 1
k wk wpk p
k tk w
k t k tk mk m k mk m k p
k wk wpk p
IUIU
ZZ
Z ZZZ ZZ Z
IUIU
(15)
where:




  













) 1 (101 00 0 1 00 1
) () (
) ( ) () () ( ) () ( ) (
) (
) () () (
) (
) 1 () 1 () 1 (
) 1 (
k tk w
k t k tk mk m k mk m k p
k
k wk wpk p
k
k wk wpk p
k
ZZ
Z ZZZ ZZ Z
E
IUIU
N
IUIU
M (16)

Figure 4. Span k between two towers

In the same way, the right-side quantities of the span no. k
could be expressed as a function of the left-side quantities of
the same span:
] [ ] [ ] [) 1 (1
) (  k k k M E N (17),
where is inverse matrix of . Recurrently appl
ying expression (14) for all the transmission line spans,
it will be obtained:
] [1
kE ] [kE] [ ] [] [ ] [ .. ] [ ] [ ] [….] [ ] [ ] [ ] [ ] [ ] [] [ ] [ ] [
) 1 () () 1 ( ) 1 ( ) 1 ( ) ( ) 1 () 1 ( ) 1 ( ) 2 ( ) 2 ( ) 2 ( ) 3 () 1 ( ) 1 ( ) 2 (
N EN E E E MN E E N E MN E M
nn n n
           
  (18)
In order to determine all the unknowns’ quantities in Fig.
3, the following boundary conditions are necessary.
At the faulted tower, on the right side of the first span, the
faulted phase conductor and the ground wire are connected
by the phase-to-ground fault, thus . ) 1 ( ) 1 (w p U U
The c
olumn vector will be: ] [) 1 (N
11
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Advances in Electrical and Computer Engineering Volume 16, Number 1, 2016




  
) 1 () 1 () 1 (
) 1 () () (
] [we w ppe w p
IZ I IIZ I I
N (19),
where represents the resultant impedance of the
lad
der network extended beyond the fault. eZ
At th
e left terminal ; therefore the
co
lumn vector will be: ) (n w p sI I I 




 
ps p n wpn p
n
IR I IIU
M) (] [
) () 1 (
) ( (20)
The expression (17) is applied to the ( n-1) span, which
contains the last tower of the transmission line (see Fig. 3)
and the following matrix equation ca be written:
] [ ] [ ] [) 1 () 1 (
) 1 ( N E Mn
n  
 (21),
where:




  
) () 1 ( ) ( ) 1 () (
) 1 () (] [
n wn t n w n wpn p
n
IZ I IIU
M (22)
] [) 1 (nE will be computed as in expression (18).
Additionally, in order to gain the necessary number of
equations, for the last span (span n) the next expressions
could be written:
0) ( ) ( ) ( ) (      n w n m p n w n w s sU Z I Z I I R (23)



   
  
) ( ) 1 ( ) 1 () 1 ( ) 1 ( ) () (
n w n w n tn t n t n ws n w p
I I IZ I UI I I
(24)
Taking into account relations (24), expression (23)
became:
0) ( ) (
) 1 ( ) 1 () ( ) 1 ( ) ( ) (
    
 
n t n wn w s n t n w n m s p
Z IZ R Z I Z R I
(25)
Considering the boundary conditions, for the case of the
fault fed from one side, all the unknowns quantities ,
, , and can be computed from
exp
ressions (21) and (25). ) (n pU
pI) 1 (wI) 1 (n wI) (n wI
B
y choosing as the given reference value, all the
currents a
re obtained as per-unit complex values referred to
the pure reference quantity of [5]. pI
pI
I
n order to obtain all the ground wires currents, in every
span, it is enough to observe that:
] [ ] [ ] [) 1 (1
) 2 ( 
  n n M E M (26), Because is known from expressions (21), by
p
roceeding toward the fault, all ground wires currents could
now be determined. ] [) 1 (nM
II
I. RESULTS
In order to illustrate and validate the theoretical
approaches outlined in the sections above, first it is
considered that the line which connects those two stations is
a 110 kV single circuit transmission line with aluminum-steel ( ACSR ) 185/32 mm
2 phase conductors and one
aluminum-steel (ACSR) 95/55 mm2 ground wire (Fig. 5).
Ground wire impedance per one span and the
m
utual impedances per one span between the ground
wire a
nd the faulted phase are calculated, for different
values of the soil resistivity ) (k wZ
) (k mZ
, with formulas based on
Carson’s theory of the ground return path [1, 21].
Line impedances per one span are determined considering
that the average length of the span is 250 m.
The fault was assumed to occur on the phase which is the
furthest from the ground wire , because the lowest coupling
between the phase and the ground wire will produce the highest tower voltage.
The total fault current was chosen as the reference
val
ue given, thus all the curre nts are obtained as per-unit
complex values referred to th e pure reference quantity of
. pI
pI

Figure 5. Disposition of line conductors

Fig. 6 shows the currents flowing in the ground wire in
the case of a fault at the last tower of the line, considering a
single circuit transmission line.
It was assumed that the line has 15 towers and tower
impedances are uniform, first 5tZ and then
10tZ. The values were computed considering the
model for uniform tower impedances (Model 1 above) and
then, the model for non-uniform tower impedances (Model 2
above).
The first model will be further called Model 1 and the
second one will be further called Model 2.
It can be seen that in case of a uniform tower resistances
the values are identical, conf irming the correctness of the
two models.
The values were obtained for soil resistivity

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Advances in Electrical and Computer Engineering Volume 16, Number 1, 2016
100 m, for the source station grounding system
resistance  001 . 0sR and for the grounding system
resistance of the distribution station  01 . 0'
sR (both
depending of soil resistivity).
Figure 6. The currents flowing in the ground wire, uniform tower
resistances
Fig. 7 shows the voltage rise of the faulted tower as a
function of the soil resistivity. It was assumed that the line
has 15 towers and tower impedances are uniform 5tZ.
The values are obtained for 01. 0sR and
 1 . 0'
sR .

Figure 7. Voltage rise of the fau lted tower as a function of the soil
resistivity

Fig. 8 and Fig. 9 are showing the currents flowing in the
ground wire in the case of a fau lt at the last tower of the line,
considering a single circuit transmission line, for
50 m, respectively for 100 m. The values
were computed using both Model 1 and Model 2. It was assumed that the line has 15 towers and tower impedances
are uniform and respectively non-uniform.
In the last case (non-uniform impedances) it was assumed
that first 7 towers have the same impedance
values 5tZ, and the last 3 towers have 10tZ.
In order to use Model 1 for this computation, we
developed two scenarios. In the first one we approximate all transmission towers’ impedances with
5tZ,
practically considering that the last 3 towers have the same
impedance as the others. Then, in the second case, it was
considered that all towers have the equivalent uniform
impedance value 07 , 6tZ . The values are obtained for
 001 . 0sR and 01 . 0'
sR .
Figure 8. Ground wire currents for uniform, respective non-uniform tower
resistances and for m50

Figure 9. wire currents for unifo rm, respective non-uniform tower
resistances and for m100

It can be seen that in this case the values are slightly
11
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Advances in Electrical and Computer Engineering Volume 16, Number 1, 2016
di
fferent, the maximum error being of 2.24%. However, it is
interesting to observe that Model 1 approximates the
calculus under the above assumptions in an acceptable
manner. Using in Model 1 the tower impedances’ average
value generates slightly more accurate results than using the minimum value.
In order to point out the influence of the stations
grounding systems on the ground fault current distribution,
Fig. 10 and Fig. 11 show the currents flowing in the ground
wire, respectively in the transmission towers, in the case of a fault at the last tower of the line, considering a single circuit
transmission line, for
50 m, respectively for
100 m
, but this time the values were obtained for
01 . 0sR and 1 .0'
sR .

Figure 10. Ground wire currents for  01 . 0sR ,  1 . 0'
sR

Figure 11. Currents flowing in transmission towers for  01 . 0sR ,
 1 . 0'
sR

It was assumed that the line has 15 towers and tower impedances are uniform ( 10tZ) and respectively non-
uniform. In the last case (non-uniform impedances) it was
assumed that first 7 towers have the same impedance values
5tZ, and the last 3 towers have 10tZ.
IV. CONCLUSIONS
Two fundamental tendencies can be observed in the
ground fault currents analysis methods: first, a continuous
effort is developed in order to make these methods more convenient for applications – considering the great number
of cases that should be solved, and second, the accuracy of these methods is improved by including new factors of lower significance. Unfortunately, some of the necessaries input data, like the footing resistance of each tower, can be
found only when the transmission line is already built. But,
the increased accuracy of thes e models has not a practical
importance for most of the problems [9, 10].
In this paper, the ground fault current distribution is
studied and evaluated for a ty pical electrical network with
overhead transmission lines. It was considered an overhead transmission line with one ground wire, connected to the ground at every tower of the lin e and the fault appears at the
terminal tower. There were pr esented the expressions of the
currents flowing to ground through the towers and the currents in every span of the ground conductor in two cases:
uniform, respectively non-uniform tower impedances and span lengths. In the first case, these currents are varying
exponentially and their expressions contain two arbitrary
parameters. For the long lines case, one of these parameters could be neglected. It was established the minimum number of towers that fulfill this request. For lines with a smaller
number of towers, both the parameters must be determined.
Based on the boundary conditions , there were obtained these
parameters’ expressions.
In the second case, a mathematical model inspired by
Sebo’s work [5] was described. The expressions of the currents flowing to ground through the towers and the currents in every span of the ground conductor could be computed taking into account the non-uniformity of the towers footing impedances, respectively of the power lines
spans lengths.
A laborious parametric analysis was done in order to
comparatively study the effects of the non-uniformity of the
towers footing impedances, respectively of the lengths of spans of power lines and soil resistivity on the ground fault
current distribution in substations, overhead ground wires and towers.
A complex MATLAB computer program fully
implementing the presented mathematical methods has been developed. The models were simulated on different realistic validation cases, generating intuitive useful results for the designer.
The tower impedances and the transmission line
impedances depend on the soil resistivity. It was studied the effect of the soil resistivity on the ground fault current distribution, by computing the impedances of the transmission line and ground wire with the formula based on
Carson equation which contains the soil resistivity; the
grounding systems of the terminal substation was also given

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Advances in Electrical and Computer Engineering Volume 16, Number 1, 2016
114 as a function of a soil resistivity. Thus, from the above
figures could be seen the strong influence of the soil resistivity on the ground fault currents distribution.
The currents which return through ground wires and
transmission towers was computed and examined for various validation cases. It was shown that the non-uniformity of the transmission tower impedances might have considerable effects on the gro und fault current distribution.
The results clearly show that ignoring the ground return currents may lead to grounding over-design.
R
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